Differential mimo transceiver

ABSTRACT

A receiver detects a plurality of information symbols. A first signal and a second signal are received from a first transmitted signal and a second transmitted signal, respectively, transmitted by a first plurality of antennas and received by a second plurality of antennas connected to the receiver. The second signal is received after the first signal. The receiver spatially filters the first signal using a spatial filter matrix. The receiver computes a conjugate of the first filtered signal to define a conjugate first signal, and spatially filters a second signal using the spatial filter matrix. The receiver computes a Hadamard product of the first filtered signal and the conjugate first signal to define a differential measurement signal. The receiver detects the plurality of information symbols from the differential measurement signal.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 14/619,612 that was filed Feb. 11, 2015, the entire contents ofwhich are hereby incorporated by reference.

REFERENCE TO GOVERNMENT RIGHTS

This invention was made with government support under 1247583 awarded bythe National Science Foundation and FA860-13-C-7351 awarded by theUSAF/ESC. The government has certain rights in the invention.

BACKGROUND

In a multiple-input, multiple-output (MIMO) system multiple antennas areused at both the transmitter and the receiver to improve communicationperformance. MIMO techniques are a key enabler for high-capacitycommunication at high frequencies, such as millimeter-wave frequencies,that are being developed for emerging 5G wireless applications.Interference between multiple spatial data streams in MIMO systems is alimiting factor that necessitates the use of interference suppression.Linear interference suppression techniques are promising due to theirsimplicity. However, they generally require coherent channel estimation,which in turn requires the availability of a phase-coherent localoscillator at the receiver. The requirement of phase coherence betweenthe transmitter and receiver is a stringent requirement at highfrequencies, adding significant cost and complexity.

SUMMARY

In an example embodiment, a non-transitory computer-readable medium isprovided having stored thereon computer-readable instructions that whenexecuted by a computing device, cause the computing device to detect aplurality of information symbols. A first signal is spatially filteredusing a spatial filter matrix to define a first filtered signal. Thefirst signal is a result of a first transmitted signal transmitted by afirst plurality of antennas and received by a second plurality ofantennas connected to the receiver. A conjugate of the first filteredsignal is computed to define a conjugate first signal. A second signalis spatially filtered using the spatial filter matrix to define a secondfiltered signal. The second signal is a result of a second transmittedsignal transmitted by the first plurality of antennas and received bythe second plurality of antennas connected to the receiver. The secondsignal is received after the first signal. A Hadamard product iscomputed of the first filtered signal and the conjugate first signal todefine a differential measurement signal. The plurality of informationsymbols are detected from the differential measurement signal.

In another example embodiment, a receiver is provided that detects aplurality of information symbols.

In yet another example embodiment, a method is provided of detecting aplurality of information symbols.

Other principal features of the disclosed subject matter will becomeapparent to those skilled in the art upon review of the followingdrawings, the detailed description, and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Illustrative embodiments of the disclosed subject matter will hereafterbe described referring to the accompanying drawings, wherein likenumerals denote like elements.

FIG. 1 depicts a communication scenario in accordance with anillustrative embodiment.

FIG. 2 depicts a transmitter and a receiver in a multiple-input,multiple-output (MIMO) system in accordance with an illustrativeembodiment.

FIG. 3 depicts a block diagram of a receiver device in accordance withan illustrative embodiment.

FIGS. 4a-4c illustrate the performance of five communication systems forthree different levels of interference.

FIGS. 5-8 depict block diagrams of receiver devices in accordance withillustrative embodiments.

FIGS. 9a-9c illustrate the performance of five communication systems forthree different levels of interference.

FIGS. 10-13 depict block diagrams of receiver devices in accordance withadditional illustrative embodiments.

DETAILED DESCRIPTION

Referring to FIG. 1, in an illustrative communication system, there is aline-of-sight (LoS) path between a first transceiver 100 and a secondtransceiver 102 that represents clear spatial channel characteristicsthough first transceiver 100 and second transceiver 102 also may belinked in a multipath environment. For example, a signal 104 transmittedby second transceiver 102 is radiated towards first transceiver 100 onthe LoS path. First transceiver 100 and second transceiver 102 supportboth the transmission and the reception of electromagnetic waves. Use ofthe terms transmitter and receiver is to describe an example functionthat can be performed by each device. For purposes of discussion, secondtransceiver 102 is denoted as a transmitting transceiver or atransmitter, and first transceiver 100 is denoted as a receivingtransceiver or receiver though each transceiver may be configured tosupport either or both functions. First transceiver 100 is illustratedas a base station of a communications system and second transceiver 102is illustrated as a communications device that communicates with thebase station such as a cell phone though this is merely forexemplification and is not intended to be limiting.

One or both of first transceiver 100 and second transceiver 102 may bemounted on moving objects such that a distance between the transceiversmay change with time. As known to a person of skill in the art, thecommunication environment between first transceiver 100 and secondtransceiver 102 may fluctuate due to changes in environmental conditionssuch as weather, due to changes in interference sources, and due tomovement between first transceiver 100 and second transceiver 102, whichmay change the multipath environment, any of which may cause afluctuation in the received signal-to-noise ratio (SNR),signal-to-interference ratio (SIR), signal to interference and noiseratio (SINR), and/or communication channel characteristics, even wherethe transmission power and other signal characteristics such asfrequency, pulsewidth, bandwidth, etc. remain unchanged.

Referring to FIG. 2, first transceiver 100 may include a plurality ofantennas 200 arranged to form an array. The array may be a uniform or anon-uniform linear array, a rectangular array, a circular array, aconformal array, etc. The plurality of antennas 200 are mounted in acommon plane. An antenna of the plurality of antennas 200 may be adipole antenna, a monopole antenna, a helical antenna, a microstripantenna, a patch antenna, a fractal antenna, a feed horn, a slotantenna, etc. An antenna spacing, denoted d_(R), may separate each ofthe plurality of antennas 200 from an adjacent antenna of the pluralityof antennas 200 in the common plane. The plurality of antennas 200 areconfigured to receive an analog signal from second transceiver 102and/or to radiate a plurality of radio waves toward second transceiver102. The first plurality of antennas 200 may include any number ofantennas where M denotes the number of antennas included in the firstplurality of antennas 200.

Second transceiver 102 may include a second plurality of antennas 202arranged to form a second array. The second array may be a uniform or anon-uniform linear array, a rectangular array, a circular array, aconformal array, etc. The second plurality of antennas 202 are mountedin a common plane. An antenna of the second plurality of antennas 202may be a dipole antenna, a monopole antenna, a helical antenna, amicrostrip antenna, a patch antenna, a fractal antenna, a feed horn, aslot antenna, etc. A second antenna spacing, denoted d_(T), may separateeach of the second plurality of antennas 202 from an adjacent antenna ofthe second plurality of antennas 202 in the common plane. The secondplurality of antennas 202 are configured to receive an analog signalfrom first transceiver 100 and/or to radiate a plurality of radio wavestoward first transceiver 100. The second plurality of antennas 202 mayinclude any number of antennas where N denotes the number of antennasincluded in the second plurality of antennas 202.

A boresight vector 204 extends from a center of the array of firsttransceiver 100 perpendicular to the common plane in which the pluralityof antennas 200 is mounted. Second transceiver 102 is located along adirection vector 206 which defines an angle 208, which may be denotedφ_(o), relative to boresight vector 204. For illustration, φ_(o)represents only the azimuth angle relative to a linear array.Alternative embodiments can be extended to two-dimensional arrays inwhich the angle φ_(o) is replaced by a pair of angles representing theazimuth angle and the elevation angle.

In an illustrative embodiment, first transceiver 100 and secondtransceiver 102 are configured to support differential communication.Differential communication is typically used when a phase-coherent localoscillator is not available at the receiver, resulting in an unknownphase offset between the transmitter and receiver, and possibly even asmall frequency offset. In a constant modulus constellation, thetransmitted symbols may be of the form s=Ae^(jφ) for some given fixed A.Let A=1 for simplicity. In a differential communication system,information is typically encoded in a phase difference Δφ between acurrent transmit symbol s=s(t) and a previous transmit symbols_(T)=s(t−T) where T is a symbol period; that is,

s=Ae^(jφ)=e^(jΔφ)s_(T); s_(T)=Ae^(jφ) _(T).   (1)

Assuming that the differential symbols Δφ are chosen randomly from asymmetric constellation, such as in a communication system that usesquadrature phase shift keying (QPSK), and are independent across time,it follows that e^(jΔφ) is zero mean and independent of s_(T). Underthese assumptions, the following can be shown:

E[s_(T)]=0; E[s]=E[e^(jφ)]E[s_(T)]=0

|s|²=|s_(T)|²=A²=1

ss_(T) *=e ^(jΔφ) |s _(T)|² ; E[ss _(T)*]=0.   (2)

This also specifies the second-order statistics of the entire sequenceof symbols, under the assumption that the starting symbol, s₀, at timezero satisfies E[s₀]=0 and E[|s₀|²]=A²=1, which is readily satisfied.The received signals r 210 and the differential measurements are

r=e ^(jφ) _(o) s+v; r _(T) =e ^(jφ) _(o) s _(T) +v _(T)   (3)

rr _(T) *=ss _(T) *+sv _(T) *+vs _(T) *+vv _(T) *=e ^(jΔφ) +w,   (4)

where v and v_(T) represent noise, and it is assumed that the unknownphase offset φ_(o) remains constant, or varies sufficiently slowly overconsecutive symbols to enable the detection of the differentiallyencoded symbols Δφ from rr*_(T) in equation (4).

Given a general n×n MIMO system where N=M=n such that second transceiver102 and first transceiver 100 both include n antennas, the twotransmitted signal vectors for the current symbol and the previoussymbol corresponding to n differential symbols can be defined asΔφ=[Δφ₁, Δφ₂, . . . , Δφ_(n)]^(T) and

$\begin{matrix}\begin{matrix}{s = \lbrack {s_{1},s_{2},\ldots \mspace{14mu},s_{n}} \rbrack^{T}} \\{{= {{s(t)} = {\lbrack {{s_{1}(t)},{s_{2}(t)},\ldots \mspace{14mu},{s_{n}(t)}} \rbrack^{T}(6)}}}\;}\end{matrix} & (5) \\\begin{matrix}{s_{T} = \lbrack {s_{1_{T}},s_{2_{T}},\ldots \mspace{14mu},s_{n_{T}}} \rbrack^{T}} \\{= {{s( {t - T} )}(8)}} \\{= {\lbrack {{s_{1}( {t - T} )},{s_{2}( {t - T} )},\ldots \mspace{14mu},{s_{n}( {t - T} )}} \rbrack^{T}(9)}}\end{matrix} & (7)\end{matrix}$

The corresponding received signals r, r_(T) may be defined similarly.The composite 2n×1 transmitted and received signal vectors may bedefined as

$\begin{matrix}{{s_{c} = \begin{bmatrix}s \\s_{T}\end{bmatrix}};\mspace{14mu} {r_{c} = {\begin{bmatrix}r \\r_{T}\end{bmatrix}.}}} & (10)\end{matrix}$

The overall MIMO system equation for the two symbol vectors and thecomposite vector is

r=Hs, r_(T)=H_(T)s_(T); r_(C)=H_(C)s_(C)   (11)

where H=H(t) and H_(T)=H(t−T) and the 2n×2n composite channel matrixH_(C) is given by

$\begin{matrix}{H_{c} = \begin{bmatrix}H & 0 \\0 & H_{T}\end{bmatrix}} & (12)\end{matrix}$

In differential communication, it is assumed that H=H_(T); that is, thechannel does not change across two symbol durations. The followingdifferential measurements are possible at the receiver

$\begin{matrix}{R_{c} = {{r_{c}r_{c}^{H}} = \begin{bmatrix}{rr}^{H} & {rr}_{T}^{H} \\{r_{T}r^{H}} & {r_{T}r_{T}^{H}}\end{bmatrix}}} & (13)\end{matrix}$

Using equation (11), the system equation for these differentialmeasurements at the receiver (without noise) is

R_(C)=r_(C)r_(C) ^(H)=H_(C)s_(C)s_(C) ^(H)H_(C) ^(H)=H_(C)Q_(C)H_(C)^(H)   (14)

where Q_(C)=s_(C)s_(C) ^(H) is of the same form as equation (13) forr_(C)r_(C) ^(H) and represents the possibilities for differentialtransmission. Using equation (12) and expanding equation (14) results in

$\begin{matrix}{R_{c} = {\begin{bmatrix}{rr}^{H} & {rr}_{T}^{H} \\{r_{T}r^{H}} & {r_{T}r_{T}^{H}}\end{bmatrix} = {\begin{bmatrix}{{Hss}^{H}H^{H}} & {{Hss}_{T}^{H}H_{T}^{H}} \\{H_{T}s_{T}s^{H}H^{H}} & {H_{T}s_{T}s_{T}^{H}H_{T}^{H}}\end{bmatrix}.}}} & (15)\end{matrix}$

The matrix relation defined by equation (15) represents a fundamentalset of equations for understanding MIMO communication and interferencesuppression under differential signaling. Another version can beobtained by vectorizing equation (14) as

z _(C)=vec(R _(C))=[H _(C) *

H _(C) ]x _(C) , x _(C)=vec(Q _(C))   (16)

where the following relation is used

vec(ADB)=[B ^(T)

A]vec(D)   (17)

where

denotes a Kronecker product. A special case of equation (17) for vectorsa and b is

vec(ab ^(H))=[b*

a]vec(I ₁)=b*

a   (18)

A sub-system of equation (15) and equation (16) can be defined as

rr_(T) ^(H)=Hss_(T)H_(T) ^(H)=Hss_(T)H^(H),   (19)

where the assumption that H=H_(T) is applied. Vectorizing equation (19)results in

z=H _(d) x; H _(d) =[H* _(T)

H]

z=vec(rr _(T) ^(H)), x=vec(ss _(T) ^(H))   (20)

where H_(d) is a differential-MIMO (D-MIMO) channel matrix. For n=2,

$\begin{matrix}{{{rr}_{T}^{H} = \begin{bmatrix}{r_{1}r_{1_{T}}^{*}} & {r_{1}r_{2_{T}}^{*}} \\{r_{2}r_{1_{T}}^{*}} & {r_{2}r_{2_{T}}^{*}}\end{bmatrix}},} & (21) \\{{{ss}_{T}^{H} = \begin{bmatrix}{s_{1}s_{1_{T}}^{*}} & {s_{1}s_{2_{T}}^{*}} \\{s_{2}s_{1_{T}}^{*}} & {s_{2}s_{2_{T}}^{*}}\end{bmatrix}},} & (22) \\{z = {{{vec}( {rr}_{T}^{H} )} = \begin{bmatrix}{r_{1}r_{1_{T}}^{*}} \\{r_{2}r_{1_{T}}^{*}} \\{r_{1}r_{2_{T}}^{*}} \\{r_{2}r_{2_{T}}^{*}}\end{bmatrix}}} & (23) \\{{x = {{{vec}( {ss}_{T}^{H} )} = \begin{bmatrix}{s_{1}s_{1_{T}}^{*}} \\{s_{2}s_{1_{T}}^{*}} \\{s_{1}s_{2_{T}}^{*}} \\{s_{2}s_{2_{T}}^{*}}\end{bmatrix}}},{and}} & (24) \\\begin{matrix}{H_{d} = {{H_{T}^{*} \otimes H} = {H^{*} \otimes H}}} \\{= {{\begin{bmatrix}h_{11}^{*} & h_{12}^{*} \\h_{21}^{*} & h_{22}^{*}\end{bmatrix} \otimes \begin{bmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{bmatrix}}(26)}} \\{= {\begin{bmatrix}{h_{11}^{*}H} & {h_{12}^{*}H} \\{h_{21}^{*}H} & {h_{22}^{*}H}\end{bmatrix}(27)}} \\{= {\begin{bmatrix}{h_{11}}^{2} & {h_{11}^{*}h_{12}} & {h_{12}^{*}h_{11}} & {h_{12}}^{2} \\{h_{11}^{*}h_{21}} & {h_{11}^{*}h_{22}} & {h_{12}^{*}h_{21}} & {h_{12}^{*}h_{22}} \\{h_{21}^{*}h_{11}} & {h_{21}^{*}h_{12}} & {h_{22}^{*}h_{11}} & {h_{22}^{*}h_{12}} \\{h_{21}}^{2} & {h_{21}^{*}h_{22}} & {h_{22}^{*}h_{21}} & {h_{22}}^{2}\end{bmatrix}.(28)}}\end{matrix} & (25)\end{matrix}$

H_(d) is full-rank, if H is full-rank, which follows from the propertiesof the Kronecker product: rank(A

B)=rank(A)rank(B). The first and last elements of z carry theinformation about the desired differential symbols, Δφ₁ and Δφ₂,contained in the first and last elements of x. The remaining elements ofz represent cross-terms that carry information about interference. Ifthere is no inter-channel inference, H is diagonal, there is nointerference in the differential system, and H_(d) is diagonal. Theoff-diagonal entries of H_(d) represent the interference between thetransmitted signals in x (see equation (24)) that corrupt the receivermeasurements in z (see equation (23)).

The noisy underlying system equations based on equation (11) can bedefined as

r=√{square root over (ρ)}Hs+v; r _(T) =√{square root over (ρ)}H _(T) s_(T) +v _(T)   (29)

rr _(T) ^(H) =ρHss _(T) ^(H) H _(T) ^(H) +√{square root over (ρ)}Hsv_(T) ^(H) +√{square root over (ρ)}vs _(T) ^(H) H _(T) ^(H) +vv _(T) ^(H)  (30)

where v˜

(0,σ²I_(n)) and v_(T)˜

(0,σ²I_(n)) represent complex Gaussian noise vectors that areindependent of each other, the signals are s and s_(T), and ρ representsa signal-to-noise ratio (SNR) for each data stream. Vectorizing equation(30) results in a noisy version of the D-MIMO system equation (20) andis defined as

z=ρH _(d) x+w   (31)

where

w=w ₁ w ₂ w ₃=vec(√{square root over (ρ)}Hsv _(T) ^(H) +√{square rootover (ρ)}vs _(T) ^(H) H _(T) ^(H) +vv _(T) ^(H)),   (32)

x=vec(ss_(T) ^(H)) is the vector of transmitted differential symbols,z=vec(rr_(T) ^(H)) is a vector of received differential signals, and wis an effective noise vector that consists of the three terms indicatedin equation (32).

A n²×n² (4×4 for the illustrative case) matrix F_(o) can be designedthat operates on the vector z to yield estimates of x in which theinterference has been suppressed:

x_(est)=F_(o)z.   (33)

F_(o) can be defined using a minimum mean squared error (MMSE)criterion, assuming knowledge of the D-MIMO channel matrix H_(d) as:

F _(o)=arg min_(F) E[∥x _(est) −x∥ ² ]=H _(d) ^(H)(ρ² H _(d) H _(d)^(H+Σ) _(w))⁻¹   (34)

where Σ=E[ww^(H)] is a covariance matrix of w, and H_(d)H_(d)^(H)=(H*_(T)H_(T) ^(T)

HH^(H)). ρ² may know a priori in some cases or may be estimated as partof channel estimation using training signals as understood by a personof skill in the art. F_(o) is a spatial filter matrix that is(n²×n²).The differentially encoded transmitted symbols in x can beestimated at the receiver by applying differential detectors,corresponding to the differential transmission scheme used, to theappropriate elements of x_(est).

To characterize the second-order statistics of x and w in equation (31),zero-mean signal constellations for the differential symbols is assumedwith different differential symbols assumed to be independent acrosstime and data streams. This results in the following second-orderstatistics for s:

E[s]=E[s_(T)]=0, E[ss_(T) ^(H)]=0   (35)

E[ss^(H)]=E[s_(T)s_(T) ^(H)]=I_(n)   (36)

which in turn results in the following second-order statistics forx=vec(ss_(T) ^(H))

$\begin{matrix}{{{E\lbrack X\rbrack} = {{E\lbrack {{vec}( {ss}_{T}^{H} )} \rbrack} = {{{vec}( {E\lbrack {ss}_{T}^{H} \rbrack} )} = 0}}}\begin{matrix}{{E\lbrack {xx}^{H} \rbrack} = {E\lbrack {{{vec}( {ss}_{T}^{H} )}{{vec}( {ss}_{T}^{H} )}^{H}} \rbrack}} \\{= {{E\lbrack {( {s_{T}^{*} \otimes s} )( {s_{T}^{T} \otimes s^{H}} )} \rbrack} = {E\lbrack {s^{*}{s_{T}^{T} \otimes {ss}^{H}}} \rbrack}}} \\{= {{{E\lbrack {s^{*}s_{T}^{T}} \rbrack} \otimes {E\lbrack {ss}^{H} \rbrack}} = {{I_{n} \otimes I_{n}} = {I_{n^{2}}.}}}}\end{matrix}} & (37)\end{matrix}$

Assuming that the signal and noise are independent, and using theassumptions on the statistics of v and v_(T), it can be shown that

$\begin{matrix}{{E\lbrack w\rbrack} = 0} & (38) \\\begin{matrix}{\Sigma_{w} = {E\lbrack {ww}^{H} \rbrack}} \\{{= {{\rho \; {\sigma^{2}( {I_{n} \otimes {HH}^{H}} )}} + {\rho \; {\sigma^{2}( {H_{T}^{*}{H_{T}^{T} \otimes I_{n}}} )}} + {\sigma^{4}I_{n^{2}}}}}\;}\end{matrix} & (39)\end{matrix}$

where the three terms in Σ_(W) in equation (39) represent the covariancematrices of the corresponding terms in equation (32), where σ² is anoise power.

The noise statistics follow from the following calculations on the jointstatistics of w₁, w₂, and w₃ in equation (32). Using equation (18),

w ₁=√{square root over (ρ)}vec(Hsv _(T) ^(H))=√{square root over (ρ)}(v_(T) *

Hs)   (40)

w ₂=√{square root over (ρ)}vec(vs _(T) ^(H) H _(T) ^(H))=√{square rootover (ρ)}(H _(T) *s _(T) *

v)   (41)

w ₃=vec(vv _(T) ^(H))=(v _(T) *

v).   (42)

The second-order statistics of {w_(i)} are

$\begin{matrix}{{E\lbrack w_{1} \rbrack} = {{\sqrt{\rho}( {{E\lbrack v_{T}^{*} \rbrack} \otimes {E\lbrack{Hs}\rbrack}} )} = 0}} & (43) \\\begin{matrix}{{E\lbrack {w_{1}w_{1}^{H}} \rbrack} = {\rho \; {E\lbrack {( {v_{T}^{*} \otimes {Hs}} )( {v_{T}^{*} \otimes {Hs}} )^{H}} \rbrack}}} \\{= {\rho \; {E\lbrack ( {v_{T}^{*}{v_{T}^{T} \otimes {Hss}^{H}}H} ) \rbrack}}} \\{= {\rho \; \sigma^{2}{{E\lbrack {v_{T}^{*}v_{T}^{T}} \rbrack} \otimes {{HE}\lbrack {ss}^{H} \rbrack}}H^{H}}} \\{= {\rho \; \sigma^{2}{I_{n} \otimes {{HH}^{H}.}}}}\end{matrix} & (44)\end{matrix}$

Similarly,

E[w₂]=E[w₃]=0   (45)

E[w ₂ w ₂ ^(H)]=ρσ²(H _(T) *H _(T) ^(H)

I _(n))   (46)

E[w₃w₃ ^(H)]=σ²I_(n)

σ²I_(n)=σ⁴I_(n) ₂ .   (47)

It can further be similarly shown that

E[w₁w₂ ^(H)]=E[w₁w₃ ^(H)]=E[w₂w₃ ^(H)]=0.   (48)

Combining the above calculations leads to the second-order statistics ofw given in equation (39).

If HH^(H) has the eigenvalue decomposition HH^(H)=U

U^(H) and H_(T)H_(T) ^(H) has the eigenvalue decomposition H_(T)H_(T)^(H)=U_(T)

_(T)U_(T) ^(H), the noise covariance matrix Σ_(W) has the eigenvaluedecomposition

Σ_(W)=(U* _(T)

U)

(U* _(T)

U)^(H)   (49)

=ρσ²(

⊕

_(T))+σ⁴ I _(n) ₂   (50)

where A⊕B=(I

A)+(B

I) is the Kronecker sum.

H_(d) can be estimated using training symbols as understood by a personof skill in the art. The estimated version of channel matrix H_(d) isplugged into equation (34) to determine spatial filter matrix F_(o). Thetraining signals can be designed in a variety of ways. The simplestapproach may be to design the transmitted signals so that only one entryof x (see equation (24)) is non-zero in each differential trainingsymbol; the corresponding column of H_(d) can then be estimated from thecorresponding received differential measurements z (see equation (23).The training symbols that correspond to this simple approach aredescribed below. For estimating the first column of H_(d), the followingmay be transmitted

$\begin{matrix}{s = {s_{T} = \begin{bmatrix}1 \\0\end{bmatrix}}} & (51)\end{matrix}$

resulting in

$\begin{matrix}{x = {\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix}.}} & (52)\end{matrix}$

For estimating the second column of H_(d), the following may betransmitted

$\begin{matrix}{{s = \begin{bmatrix}0 \\1\end{bmatrix}},{s_{T} = \begin{bmatrix}1 \\0\end{bmatrix}}} & (53)\end{matrix}$

resulting in

$\begin{matrix}{x = {\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix}.}} & (54)\end{matrix}$

For estimating the third column of H_(d), the following may betransmitted

$\begin{matrix}{{s = \begin{bmatrix}1 \\0\end{bmatrix}},{s_{T} = \begin{bmatrix}0 \\1\end{bmatrix}}} & (55)\end{matrix}$

resulting in

$\begin{matrix}{x = {\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix}.}} & (56)\end{matrix}$

Finally, for estimating the fourth column of H_(d), the following may betransmitted

$\begin{matrix}{s = {s_{T} = \begin{bmatrix}0 \\1\end{bmatrix}}} & (57)\end{matrix}$

resulting in

$\begin{matrix}{x = {\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}.}} & (58)\end{matrix}$

From equation (39) estimates of HH^(H) and H*_(T)H_(T) ^(T) are used toestimate Σ_(W) for F_(o) in equation (34). For a case of interest,H_(T)=H, so that

vec(HH ^(H))=[H*

H]vec(I)=H _(d)vec(I).   (59)

As a result, the two matrices HH^(H) and H*_(T)H_(T) ^(T) can beextracted from H_(d).

Referring to FIG. 3, a block diagram of a receiver 300 that may beimplemented at first transceiver 100 acting as a receiving transceiveris shown in accordance with an illustrative embodiment. Secondtransceiver 102 may include similar elements as understood by a personof skill in the art. Receiver 300 may be implemented to yield estimatesof x, an estimate vector x_(est), in which the interference has beensuppressed by defining matrix F_(o) that operates on the vector z asdefined in equation (33). Receiver 300 may include a sample and holdoperator 302, a conjugate operator 304, a Kronecker product operator306, a spatial filter operator 308, and a symbol detector operator 310.Fewer, different, and additional components may be incorporated intoreceiver 300. For example, sample and hold operator 302 may beimplemented with a delay line in an analog implementation rather thanspecifically a sample and hold circuit.

Sample and hold operator 302, conjugate operator 304, Kronecker productoperator 306, spatial filter operator 308 and/or symbol detectoroperator 310 perform operations on a received signal r 210 to detect thedifferentially encoded symbol vector Δφ from rr*_(T). One or more ofsample and hold operator 302, conjugate operator 304, Kronecker productoperator 306, spatial filter operator 308 and/or symbol detectoroperator 310 may be implemented by a special purpose computer, logiccircuits, or hardware circuits as understood by a person of skill in theart. Thus, one or more of the operators may be implemented usinghardware, firmware, software, or any combination of these methods,depending on the stage at which the signal is converted from analog todigital form. Furthermore, some of these operators may be implemented inanalog (passband) domain, or in the baseband domain. For example, one ormore of sample and hold operator 302, conjugate operator 304, Kroneckerproduct operator 306, spatial filter operator 308 and/or symbol detectoroperator 310 may be implemented in software (comprised ofcomputer-readable and/or computer-executable instructions) stored in acomputer-readable medium and accessible by a processor for execution ofthe instructions that embody the operations of the associated operator.The instructions may be written using one or more programming languages,assembly languages, scripting languages, etc.

A computer-readable medium is an electronic holding place or storage forinformation so the information can be accessed by the processor asunderstood by those skilled in the art. The computer-readable medium caninclude, but is not limited to, any type of random access memory (RAM),any type of read only memory (ROM), any type of flash memory, etc. suchas magnetic storage devices (e.g., hard disk, floppy disk, magneticstrips, . . . ), optical disks (e.g., compact disc (CD), digitalversatile disc (DVD), . . . ), smart cards, flash memory devices, etc.Controller 102 may have one or more computer-readable media that use thesame or a different memory media technology. Receiver 300 may includeone or more computer-readable media.

A processor performs operations as understood by those skilled in theart. A digital signal processor (DSP) is a type of processor thatoperates on digital signals. The processor may be implemented inhardware and/or firmware. The processor may execute an instruction,meaning the processor performs/controls the operations called for bythat instruction. The term “execution” is the process of running anapplication or the carrying out of the operation called for by aninstruction. The processor operably couples with the computer-readablemedium to read, to store, and to process information. The processor mayretrieve a set of instructions from a permanent memory device and copythe instructions in an executable form to a temporary memory device thatis generally some form of RAM. Receiver 300 may include a plurality ofprocessors that use the same or a different processing technology.

Sample and hold operator 302 samples and holds a copy of a previouslyreceived signal r_(T) 314. Conjugate operator 304 may compute a complexconjugate of the sampled and held previously received signal r_(T) 314as a conjugate signal, r*_(T) 316. Kronecker product operator 306computes a differential measurement signal z 318, z=vec(rr_(T)^(H))=r_(T)

r. Spatial filter operator 308 operates on differential measurementsignal z 318 to yield estimates of x, an estimate vector x_(est) 320, inwhich the interference has been suppressed by applying equation (33),x_(est)=F_(o)z. As discussed above, F_(o) is a spatial filter matrixthat can be defined using equation (34) based on knowledge of theestimated D-MIMO channel matrix H_(d), which can be estimated usingtraining symbols as discussed above with reference to equations(51)-(58). Symbol detector operator 310 detects the differentiallyencoded symbols Δφ 322, for example, by applying differential detectorsto the estimates of x, estimate vector x_(est) 320.

To illustrate the performance, probability of error P_(e) versus SNR forreceiver 300 implemented as an n×n MIMO system with n=2 antennas wascalculated based on uncoded QPSK differential transmissions. The P_(e)was computed numerically from 1,000,000 symbols, and the phases of theentries of H were changed randomly every 1,000 symbols. Five differentreceiver systems were simulated: 1) receiver 300 based on estimatingchannel matrix H_(d) using training symbols at the same SNR as that fordata communication, 2) receiver 300 assuming perfect channel stateinformation, e.g., perfect knowledge of H_(d), 3) a receiver withoutinterference suppression (F_(o)=l_(n)2), 4) a coherent systemcorresponding to two non-interfering QPSK data streams, and 5) acorresponding differential system. FIGS. 4a-4c illustrate theperformance of the five systems for 3 different levels of interference.In FIG. 4 a, the interference is strongest: |h₁₂|² and |h₂₁|² are 3decibels (dB) below |h₁₁|²=|h₂₂|². In FIG. 4 b, the interference is 6 dBbelow the signal. In FIG. 4 c, the interference is 10 dB below thesignal.

Referring to FIG. 4 a, a first curve 400 shows the results for receiverconfiguration 1); a second curve 402 shows the results for receiverconfiguration 2); a third curve 404 shows the results for receiverconfiguration 3); a fourth curve 406 shows the results for receiverconfiguration 4); and a fifth curve 408 shows the results for receiverconfiguration 5). Referring to FIG. 4 b, a first curve 410 shows theresults for receiver configuration 1); a second curve 412 shows theresults for receiver configuration 2); a third curve 414 shows theresults for receiver configuration 3); a fourth curve 416 shows theresults for receiver configuration 4); and a fifth curve 418 shows theresults for receiver configuration 5). Referring to FIG. 4 c, a firstcurve 420 shows the results for receiver configuration 1); a secondcurve 422 shows the results for receiver configuration 2); a third curve424 shows the results for receiver configuration 3); a fourth curve 426shows the results for receiver configuration 4); and a fifth curve 428shows the results for receiver configuration 5). The coherent system ofreceiver configuration 4) exhibited the best performance. Receiverconfiguration 5), the differential system, had a 3 dB loss compared tothe coherent system of receiver configuration 4). Receiver configuration2) exhibited the next best performance relative to receiverconfiguration 5). Receiver configuration 1) exhibited the next bestperformance relative to receiver configuration 2). The worst performanceis that of receiver configuration 3) without interference suppression.Receiver configurations 1) and 2) provide very competitive performance,whereas ignoring interference can result in unacceptably high P_(e).Receiver configurations 4 and 5 are idealized configurationscorresponding to an interference-free system for comparison.

Referring to FIG. 5, a block diagram of a second receiver 500 that maybe implemented at first transceiver 100 acting as a receivingtransceiver is shown in accordance with an illustrative embodiment.Second transceiver 102 may include similar elements as understood by aperson of skill in the art. Similar to receiver 300, second receiver 500may be implemented to yield estimates of x, estimate vector x_(est), inwhich the interference has been suppressed by defining spatial matrixF_(o) that operates on differential measurement signal z as defined inequation (33). Second receiver 500 illustrates a completely digitalimplementation of receiver 300 where all of the receiver operations areperformed using a DSP 510. Second receiver 500 may include a localoscillator 502, a mixer 504, an analog-to-digital converter (ADC) 506,sample and hold operator 302, conjugate operator 304, Kronecker productoperator 306, spatial filter operator 308, symbol detector operator 310.Fewer, different, and additional components may be incorporated intosecond receiver 500.

Received signal r 210 is mixed with a local oscillator signal 503generated by local oscillator 502 to form a mixed signal 505. Localoscillator 502 and mixer 504 downmix the received passband signal,received signal r 210, to baseband. Mixed signal 505 is input to ADC506, which converts mixed signal 505 to a digital, baseband signal r210′. Sample and hold operator 302, conjugate operator 304, Kroneckerproduct operator 306, spatial filter operator 308 and symbol detectoroperator 310 are configured to operate on the digital, baseband versionof received signal r 210.

Again, sample and hold operator 302 samples and holds a copy of apreviously received digital signal r_(T) 314′. Conjugate operator 304computes a complex conjugate of the sampled and held previously receiveddigital signal signal r 314′ as digital conjugate signal r*_(T) 316′.Kronecker product operator 306 computes a digital, differentialmeasurement signal z 318′. Spatial filter operator 308 operates ondigital, differential measurement signal z 318′ to yield digitalestimates of x, a digital estimate vector x_(est) 320′, in which theinterference has been suppressed by applying equation (33),x_(est)=F_(o)z. Symbol detector operator 310 detects the differentiallyencoded symbols Δφ 322 from digital estimate vector x_(est) 320′.

Second receiver 500 further may include a switch 508, a differentialchannel estimation operator 512, and a spatial filter calculationoperator 514. Switch 508, differential channel estimation operator 512,and spatial filter calculation operator 514 may also be implementedusing DSP 510. A position of switch 508 depends on whether secondreceiver 500 is in a channel estimation phase or a data communicationphase. The data communication phase is illustrated in FIG. 5 based onthe position of switch 508. In the channel estimation phase, trainingdata, for example, as discussed with reference to equations (51)-(58),is received. In the channel estimation phase, digital, differentialmeasurement signal z 318′ generated by the training data, is provided todifferential channel estimation operator 512, which computes an estimateof channel matrix H_(d) 516 input to spatial filter calculation operator514. Spatial filter calculation operator 514 computes spatial filtermatrix F_(o) 518 using equation (34) based on estimated channel matrixH_(d) 516 and provides F_(o) 518 to spatial filter operator 308 for usein the data communication phase.

Referring to FIG. 6, a block diagram of a third receiver 600 that may beimplemented at first transceiver 100 acting as a receiving transceiveris shown in accordance with an illustrative embodiment. Secondtransceiver 102 may include similar elements as understood by a personof skill in the art. Similar to receiver 300, third receiver 600 may beimplemented to yield estimate vector x_(est), in which the interferencehas been suppressed by defining spatial filter matrix F_(o) thatoperates on differential measurement signal z 318 as defined in equation(33). Third receiver 600 illustrates an implementation in which thedifferential measurements, z=vec(rr_(T) ^(H))=r_(T)*

r, are performed using analog passband devices. Local oscillator 502 andmixer 504 are not needed. Third receiver 600 may include sample and holdoperator 302, conjugate operator 304, Kronecker product operator 306,ADC 506, switch 508, spatial filter operator 308, symbol detectoroperator 310, differential channel estimation operator 512, and spatialfilter calculation operator 514. Fewer, different, and additionalcomponents may be incorporated into third receiver 600.

Differential measurement signal z 318 is input to ADC 605, whichconverts signal z 318 to digital, differential measurement vector z318′. The remaining receiver operators, spatial filter operator 308,symbol detector operator 310, differential channel estimation operator512, and spatial filter calculation operator 514 are implemented usingDSP 510 similar to second receiver 500.

The data communication phase is illustrated in FIG. 6 based on theposition of switch 508. In the channel estimation phase, training data,for example, as discussed with reference to equations (51)-(58), isreceived. Switch 508 is positioned to switch digital, differentialmeasurement signal z 318′ output from ADC 506 to differential channelestimation operator 512 during the channel estimation phase. Similar tosecond receiver 500, in the channel estimation phase, digital,differential measurement signal z 318′ generated by the training data,is provided to differential channel estimation operator 512, whichcomputes an estimate of channel matrix H_(d) 516 input to spatial filtercalculation operator 514. Spatial filter calculation operator 514computes spatial filter matrix F_(o) 518 using equation (34) based onestimated channel matrix H_(d) 516 and provides F_(o) 518 to spatialfilter operator 308 for use in the data communication phase.

Referring to FIG. 7, a block diagram of a fourth receiver 700 that maybe implemented at first transceiver 100 acting as a receivingtransceiver is shown in accordance with an illustrative embodiment.Second transceiver 102 may include similar elements as understood by aperson of skill in the art. Similar to receiver 300, fourth receiver 700may be implemented to yield estimate vector x_(est), in which theinterference has been suppressed by defining spatial filter matrix F_(o)that operates on differential measurement signal z 318 as defined inequation (33).

Fourth receiver 700 illustrates an implementation in which thedifferential measurements and the spatial filtering are performed withanalog passband and baseband devices, respectively. Again, localoscillator 502 and mixer 504 are not needed. Fourth receiver 700 mayinclude sample and hold operator 302, conjugate operator 304, Kroneckerproduct operator 306, switch 506, spatial filter operator 308, ADC 506,symbol detector operator 310, differential channel estimation operator512, spatial filter calculation operator 514, a second switch 702, athird switch 704, and a digital-to-analog converter (DAC) 706. Fewer,different, and additional components may be incorporated into fourthreceiver 700.

Estimate vector x_(est) 320 is input to ADC 506, which converts estimatevector x_(est) 320 to digital estimate vector x_(est) 320′ that is inputto symbol detector operator 310. The remaining receiver operators,symbol detector operator 310, differential channel estimation operator512, and spatial filter calculation operator 514 are performed using DSP510 similar to second receiver 500. Switch 508 is positioned to switchdifferential measurement signal z 318 between spatial filter operator308 and differential channel estimation operator 512. Second switch 702is positioned between spatial filter operator 308 and ADC 506. Thirdswitch 704 is positioned between ADC 506 and symbol detector operator310. Switch 508, second switch 702, and third switch 704 switchsimultaneously so that, in the channel estimation phase, differentialmeasurement signal z 318 is input to ADC 506 and digital, differentialmeasurement signal z 318′ is input to differential channel estimationoperator 512, which computes the estimate of channel matrix H_(d) 516input to spatial filter calculation operator 514. Spatial filtercalculation operator 514 computes spatial filter matrix F_(o) 518 usingequation (34) based on channel matrix H_(d) 516 and provides spatialfilter matrix F_(o) 518 to DAC 706, which generates an analog, basebandspatial filter matrix F_(o) 518′ to spatial filter operator 308 for usein the data communication phase.

A quasi-coherent estimate of a second channel matrix H can be obtainedfrom channel matrix H_(d) and can be used for linear interferencesuppression on direct receiver measurements r and r_(T), rather than onz=vec(rr_(T) ^(H)) followed by differential detection from appropriateelements of z. The following channel decomposition of second channelmatrix H can be defined

H=H_(o)

_(φ)  (60)

where H is the actual channel matrix

$\begin{matrix}{{H = \begin{bmatrix}{{h_{11}}e^{j\; \angle \; h_{11}}} & {{h_{12}}e^{j\; \angle \; h_{12}}} \\{{h_{21}}e^{j\; \angle \; h_{21}}} & {{h_{22}}e^{j\; \angle \; h_{22}}}\end{bmatrix}},} & (61)\end{matrix}$

A third channel matrix H_(o) is what can be estimated from channelmatrix H_(d) as defined below

$\begin{matrix}{{H_{O} = \begin{bmatrix}{h_{11}} & {{h_{12}}e^{j\; {({{\angle \; h_{12}} - {\angle \; h_{22}}})}}} \\{{h_{21}}e^{j\; {({{\angle \; h_{21}} - {\angle \; h_{11}}})}}} & {h_{22}}\end{bmatrix}},} & (62)\end{matrix}$

and

_(φ) is a diagonal matrix defined as

_(φ)=diag(e^(i∠h) ¹¹ , e^(j∠h) ²² ). H_(o) can be estimated from H_(d)based on equation (27). The first column of h₁₁*H/|h₁₁| yields the firstcolumn of third channel matrix H_(o). Similarly, the second column ofh₂₂*H/|h₁₁| yields the second column of third channel matrix H_(o).Thus, when using the simple channel estimation approach described byequations (51) and (57), in the quasi-coherent case only the trainingsymbols in equations (51) and (57) are needed to estimate the first andfourth columns of channel matrix H_(d) needed to determine third channelmatrix H_(o).

An MMSE filter matrix F is defined by

F=H ^(H)(ρHH ^(H)+σ² I _(n))⁻¹=

_(φ) ^(H) H _(o) ^(H)(ρH _(o) H _(o) ^(H)+σ² I _(n))⁻¹=

_(φ) ^(H) F _(o),    (63)

which operates on the baseband signal vector r. A second spatial filtermatrix F_(o) in equation (6) can be computed at the receiver and usedfor interference suppression. Thus, the processed signal vector fromwhich the differentially encoded symbols are detected can be defined by

y=F _(o) r=F _(o) Hs+F _(o) v,   (64)

The use of second spatial filter matrix F_(o), rather than MMSE filtermatrix F, does not impact the ability to detect differential symbolssince the i-th differentially encoded transmitted symbol in s_(i)s_(iT)*is detected from the product y_(i)y_(iT)*, which corresponds todetecting the differentially encoded symbol vector via y°y_(T)* where °denotes the Hadamard (element-wise) product. Second spatial filtermatrix F_(o) has order (n×n) rather than the order (n²×n²) of spatialfilter matrix F_(o) defined for receivers 300, 500, 600, and 700.

Interference suppression using precoding at the transmitter is alsopossible. In reciprocal channels, if the transmitter first acts as areceiver and estimates the channel matrix from differential measurementsbased on training symbols from the receiver, the following decompositionof second channel matrix H results

H=

_(φ)H_(o),   (65)

In this case, the transmitted signal may be precoded as s→Gs_(V) where

G=αF, α=√{square root over (ρ/tr(FΛ _(S) F ^(H)))}

F=(H ^(H) H+ζI)⁻¹ H ^(H), ζ=σ²/ρ,   (66)

where s_(V) is the symbol vector, ρ represents transmit power (SNR ifσ²=1) per data stream, and Λ=E[ss^(H)] is a diagonal covariance matrixof transmitted symbols, which is Λ_(S)=I, and where tr(A) denotes thetrace of a square matrix A, which is the sum of the diagonal entries ofA. The composite system matrix with precoding can be defined as

r=HGs+v   (67)

and the composite matrix HG controls the interference. In terms of thirdchannel matrix H_(o), F is defined by

F=(H _(o) ^(H) H _(o+ζI))⁻¹ H _(o) ^(H)Λ_(φ) *=F _(o)Λ_(φ)*   (68)

where second spatial filter matrix F_(o) can be computed based on thirdchannel matrix H_(o). The unknown phases in Λ_(φ)* are inconsequentialfrom the viewpoint of differential signaling, and the receiver candirectly detect the symbols differentially from z=vec(rr_(T) ^(H))because interference suppression is performed at the transmitter.

Referring to FIG. 8, a block diagram of a fifth receiver 800 that may beimplemented at first transceiver 100 acting as a receiving transceiveris shown in accordance with an illustrative embodiment. Fifth receiver800 may be referred to as an example of a quasi-coherent receiver thatsuppresses interference. Second transceiver 102 may include similarelements as understood by a person of skill in the art. Fifth receiver800 may be implemented to define second spatial filter matrix F_(o) thatoperates on received signal r 210 as defined in equation (64). Fifthreceiver 800 may include a second spatial filter operator 801, a secondsample and hold operator 802, a second conjugate operator 804, aHadamard product operator 806, and symbol detector operator 310. Fewer,different, and additional components may be incorporated into fifthreceiver 800.

Similar to sample and hold operator 302, conjugate operator 304,Kronecker product operator 306, and spatial filter operator 308, secondspatial filter operator 801, second sample and hold operator 802, secondconjugate operator 804, and Hadamard product operator 806 may beimplemented by a special purpose computer, logic circuits, or hardwarecircuits (analog or digital) as understood by a person of skill in theart. Thus, second spatial filter operator 801, second sample and holdoperator 802, second conjugate operator 804, and Hadamard productoperator 806 may be implemented using hardware, firmware, software, orany combination of these methods. For example, second spatial filteroperator 801, second sample and hold operator 802, second conjugateoperator 804, and Hadamard product operator 806 may be implemented insoftware (comprised of computer-readable and/or computer-executableinstructions) stored in a computer-readable medium and accessible by aprocessor for execution of the instructions that embody the operationsof the associated operator. The instructions may be written using one ormore programming languages, assembly languages, scripting languages,etc. Fifth receiver 800 may include one or more computer-readable media.Fifth receiver 800 may include a plurality of processors that use thesame or a different processing technology.

Both Hadamard product operator 806 and Kronecker product operator 306generate differential measurements between a current and a previousmeasurement. Hadamard product operator 802 performs operations y°y_(T)*where ° denotes the Hadamard element-wise product.

Second spatial filter operator 801 operates on the received signal r 210to yield y=F_(o)r=F_(o)Hs+F_(o)v. Second spatial filter matrix F_(o) canbe defined using equation (63) based on an estimate of third channelmatrix H_(o) determined using equation (62) based on an estimate ofchannel matrix H_(d), which can be estimated using training symbols asdiscussed above with reference to equations (51)-(58). Second sample andhold operator 802 samples and holds a copy of a previously filteredsignal y_(T) 808. Second conjugate operator 804 computes a complexconjugate of the sampled and held previously filtered signal y_(T) 810as conjugate signal y*_(T) 812. Hadamard product operator 806 computesdifferential measurement signal y°y_(T)* 814. Symbol detector operator310 detects the differentially encoded symbols Δφ 322 from differentialmeasurement signal y°y_(T)* 814, for example, by applying differentialdetectors.

To illustrate the performance, probability of error P_(e) versus SNR forfifth receiver 800 implemented as an n×n MIMO system with n=2 antennaswas calculated based on uncoded QPSK differential transmissions. TheP_(e) was computed numerically from 1,000,000 symbols, and the phases ofthe entries of H were changed randomly every 1,000 symbols. Fivedifferent receiver systems were simulated: 1) fifth receiver 800 basedon estimating H_(d) using training symbols at the same SNR as that fordata communication, 2) fifth receiver 800 assuming perfect channel stateinformation, e.g., perfect knowledge of H_(d), 3) a receiver withoutinterference suppression (F_(o)=I_(n) ₂ ), 4) a coherent systemcorresponding to two non-interfering QPSK data streams, and 5) acorresponding differential system. FIGS. 9a-9c illustrate theperformance of the five systems for 3 different levels of interference.

In FIG. 9 a, the interference is strongest: |h₁₂|² and |h₂₁|² are 3decibels (dB) below |h₁₁|²=|h₂₂|². In FIG. 9 b, the interference is 6 dBbelow the signal. In FIG. 9 c, the interference is 10 dB below thesignal. Referring to FIG. 9 a, a first curve 900 shows the results forreceiver configuration 1); a second curve 902 shows the results forreceiver configuration 2); a third curve 904 shows the results forreceiver configuration 3); a fourth curve 906 shows the results forreceiver configuration 4); and a fifth curve 908 shows the results forreceiver configuration 5). Referring to FIG. 9 b, a first curve 910shows the results for receiver configuration 1); a second curve 912shows the results for receiver configuration 2); a third curve 914 showsthe results for receiver configuration 3); a fourth curve 916 shows theresults for receiver configuration 4); and a fifth curve 918 shows theresults for receiver configuration 5). Referring to FIG. 9 c, a firstcurve 920 shows the results for receiver configuration 1); a secondcurve 922 shows the results for receiver configuration 2); a third curve924 shows the results for receiver configuration 3); a fourth curve 926shows the results for receiver configuration 4); and a fifth curve 928shows the results for receiver configuration 5). The coherent system ofreceiver configuration 4) exhibited the best performance. Receiverconfiguration 5), the differential system, had a 3 dB loss compared tothe coherent system of receiver configuration 4). Receiver configuration2) exhibited the next best performance relative to receiverconfiguration 5). Receiver configuration 1) exhibited the next bestperformance relative to receiver configuration 2). The worst performanceis that of receiver configuration 3) without interference suppression.Receiver configurations 1) and 2) provide very competitive performance,and are comparable to receiver configurations 1) and 2) using receiver300. Receiver configuration 1) using receiver 300 performs slightlyworse than receiver configuration 1) using fifth receiver 800.

FIGS. 10-13 show implementations of a quasi-coherent MIMO receiver basedon fifth receiver 800. Because the quasi-coherent MIMO receiver usescross-channel and co-channel differential measurements during thechannel estimation phase and only co-channel differential measurementsduring the data communication phase, the Hadamard product is computedduring the data communication phase and the Kronecker product iscomputed during the channel estimation phase.

Referring to FIG. 10, a block diagram of a sixth receiver 1000 that maybe implemented at first transceiver 100 acting as a receivingtransceiver is shown in accordance with an illustrative embodiment.Second transceiver 102 may include similar elements as understood by aperson of skill in the art. Similar to fifth receiver 800, sixthreceiver 1000 may be implemented to define second spatial filter matrixF_(o) that operates on received signal r 210 as defined in equation(64). Sixth receiver 1000 illustrates a completely digitalimplementation where all of the receiver operations are performed usingDSP 510.

Similar to second receiver 500, sixth receiver 1000 may include localoscillator 502, mixer 504, ADC 506, and switch 508. Received signal r210 is mixed with local oscillator signal 503 generated by localoscillator 502 to form mixed signal 505. Local oscillator 502 and mixer504 downmix the received passband signal, received signal r 210, tobaseband. Mixed signal 505 is input to ADC 506, which converts mixedsignal 505 to digital, baseband signal r 210′. Similar to secondreceiver 500, sixth receiver 1000 further may include switch 508 that isswitched between the data communication phase shown in FIG. 10 and thechannel estimation phase.

Similar to fifth receiver 800, sixth receiver 1000 may include secondspatial filter operator 801, second sample and hold operator 802, secondconjugate operator 804, Hadamard product operator 806, and symboldetector operator 310 used in the data communication phase. Each ofsecond spatial filter operator 801, second sample and hold operator 802,second conjugate operator 804, Hadamard product operator 806, and symboldetector operator 310 is implemented using DSP 510 and connected asdiscussed with reference to FIG. 8 to receive and process digital,baseband signal r 210′ to detect the differentially encoded symbols Δφ322. The signals processed by second spatial filter operator 801, secondsample and hold operator 802, second conjugate operator 804, Hadamardproduct operator 806, and symbol detector operator 310 are configured tooperate on digital signals.

Similar to second receiver 500, sixth receiver 1000 further may includesample and hold operator 302, conjugate operator 304, Kronecker productoperator 306 connected to receive and process digital, baseband signal r210′. To support the channel estimation phase, sixth receiver 1000further may include a quasi-coherent channel estimation operator 1002and a second spatial filter computational operator 1004. Quasi-coherentchannel estimation operator 1002 generates an estimate of third channelmatrix H_(o) 1006 from a computation of channel matrix H_(d) based onequation (27). For example, quasi-coherent channel estimation operator1002 first implements differential channel estimation operator 512 toestimate channel matrix H_(d) 516 and then estimates third channelmatrix H_(o) 1006 from channel matrix H_(d) 516. Second spatial filtercomputational operator 1004 computes second spatial filter matrix F_(o)1008 in digital form from third channel matrix H_(o) 1006 using equation(63) and provides the computation to second spatial filter operator 801.Fewer, different, and additional components may be incorporated intosixth receiver 1000.

Referring to FIG. 11, a block diagram of a seventh receiver 1100 thatmay be implemented at first transceiver 100 acting as a receivingtransceiver is shown in accordance with an illustrative embodiment.Second transceiver 102 may include similar elements as understood by aperson of skill in the art. Similar to fifth receiver 800, seventhreceiver 1100 may be implemented to define second spatial filter matrixF_(o) that operates on received signal r 210 as defined in equation(64). Seventh receiver 1100 also illustrates a completely digitalimplementation where all of the receiver operations are performed usingDSP 510.

Seventh receiver 1100 differs from sixth receiver 1000 in that seventhreceiver 1100 includes a configurable product operator 1102, secondswitch 702, and third switch 704. Second switch 702 is positioned on aninput side of configurable product operator 1102, and third switch 704is positioned on an output side of configurable product operator 1102.Switch 508, second switch 702, and third switch 704 are switchedsimultaneously to switch between the data communication phase shown inFIG. 11 and the channel estimation phase. Configurable product operator1102 is configured to perform the Hadamard product in the datacommunication phase and to perform the Kronecker product in the channelestimation phase. As a result, the inputs to configurable productoperator 1102 are switched between the digital filtered inputs 808′,812′ provided to compute the Hadamard product and the digital unfilteredinputs 210′ and 316′ provided to compute the Kronecker product, and theoutputs from configurable product operator 1102 are switched betweendigital, differential measurement signal y°y_(T)* 814′ and digital,differential measurement signal z 318′. Digital, differentialmeasurement signal y°y_(T)* 814′ is input to symbol detector 310.Digital, differential measurement signal z 318′ is input toquasi-coherent channel estimation operator 1002.

Referring to FIG. 12, a block diagram of an eighth receiver 1200 thatmay be implemented at first transceiver 100 acting as a receivingtransceiver is shown in accordance with an illustrative embodiment.Second transceiver 102 may include similar elements as understood by aperson of skill in the art. Similar to fifth receiver 800, eighthreceiver 1200 may be implemented to define second spatial filter matrixF_(o) that operates on received signal r 210 as defined in equation(64). Eighth receiver 1200 illustrates an implementation similar tofourth receiver 700 in that the spatial filtering and differentialmeasurements are performed by analog devices, and the channelestimation, spatial filter computation, and symbol detection areperformed by digital devices.

Referring to FIG. 13, a block diagram of a ninth receiver 1300 that maybe implemented at first transceiver 100 acting as a receivingtransceiver is shown in accordance with an illustrative embodiment.Second transceiver 102 may include similar elements as understood by aperson of skill in the art. Similar to fifth receiver 800, may beimplemented to define second spatial filter matrix F_(o) that operateson received signal r 210 as defined in equation (64). Ninth receiver1300 also illustrates an implementation similar to fourth receiver 700in that the spatial filtering and differential measurements areperformed by analog devices, and the channel estimation, spatial filtercomputation, and symbol detection are performed by digital devices.Ninth receiver 1300 differs from eighth receiver 1200 in thatconfigurable product operator 1102 replaces separate Kronecker productoperator 306 and Hadamard product operator 806. The inputs toconfigurable product operator 1102 are switched between the filteredinputs 808, 812 provided to compute the Hadamard product and theunfiltered inputs 210 and 316 provided to compute the Kronecker product,and the outputs from configurable product operator 1102 are switchedbetween digital, differential measurement signal y°y_(T)* 814′ anddigital, differential measurement signal z 318′. Digital, differentialmeasurement signal y°y_(T)* 814′ is input to symbol detector 310.Digital, differential measurement signal z 318′ is input toquasi-coherent channel estimation operator 1002.

FIGS. 10 and 12 show implementations where Hadamard product operator 802and Kronecker product operator 306 are implemented separately. FIGS. 11and 13 show implementations where Hadamard product operator 802 andKronecker product operator 306 are implemented by configurable productoperator 1102 that can be switched between full and reduced differentialmeasurements based on the channel estimation phase or the datacommunication phase. FIGS. 10 and 11 illustrate completely digitalimplementations where all of the receiver operations are performed usingDSP 510. FIGS. 12 and 13 illustrate receivers where the spatialfiltering and differential measurements used for the channel estimationphase and the data communication phase are obtained using analog devicesimplemented in passband. In FIGS. 12 and 13, second spatial filteroperator 801 is implemented in passband; whereas in FIG. 7, spatialfilter operator 308 is implemented in baseband.

Selection between receivers 300, 500, 600, 700, 800, 1000, 1100, 1200,and 1300 depends on the system in which the receiver is beingimplemented. For existing MIMO systems equipped with local oscillatorsfor downmixing the signal to baseband before analog to digitalconversion, receivers 1000 and 1100 may be preferred due to the lowerdimension (n vs n²) of second spatial filter operator 801 versus spatialfilter operator 308 and of quasi-coherent channel estimation operator1002, which reduces the computational complexity. For a receiver thathas no local oscillator, third receiver 600 may be preferred over eighthreceiver 1200 and ninth receiver 1300 to avoid implementation of secondspatial filter operator 801 in passband, which may offset the increasedcomputational complexity of third receiver 600.

The word “illustrative” is used herein to mean serving as an example,instance, or illustration. Any aspect or design described herein as“illustrative” is not necessarily to be construed as preferred oradvantageous over other aspects or designs. Further, for the purposes ofthis disclosure and unless otherwise specified, “a” or “an” means “oneor more”. Still further, in the detailed description, using “and” or“or” is intended to include “and/or” unless specifically indicatedotherwise.

The foregoing description of illustrative embodiments of the disclosedsubject matter has been presented for purposes of illustration and ofdescription. It is not intended to be exhaustive or to limit thedisclosed subject matter to the precise form disclosed, andmodifications and variations are possible in light of the aboveteachings or may be acquired from practice of the disclosed subjectmatter. The embodiments were chosen and described in order to explainthe principles of the disclosed subject matter and as practicalapplications of the disclosed subject matter to enable one skilled inthe art to utilize the disclosed subject matter in various embodimentsand with various modifications as suited to the particular usecontemplated.

What is claimed is:
 1. A non-transitory computer-readable medium havingstored thereon computer-readable instructions that when executed by acomputing device cause the computing device to: spatially filter a firstsignal using a spatial filter matrix to define a first filtered signal,wherein the first signal is a result of a first transmitted signaltransmitted by a first plurality of antennas and received by a secondplurality of antennas connected to the receiver; compute a conjugate ofthe first filtered signal to define a conjugate first signal; spatiallyfilter a second signal using the spatial filter matrix to define asecond filtered signal, wherein the second signal is a result of asecond transmitted signal transmitted by the first plurality of antennasand received by the second plurality of antennas connected to thereceiver, wherein the second signal is received after the first signal;compute a Hadamard product of the first filtered signal and theconjugate first signal to define a differential measurement signal; anddetect a plurality of information symbols from the differentialmeasurement signal.
 2. The non-transitory computer-readable medium ofclaim 1, wherein the spatial filter matrix is computed from an estimatedchannel matrix.
 3. The non-transitory computer-readable medium of claim1, wherein the estimated channel matrix is computed by transmitting aplurality of pairs of training signals before the first signal isreceived.
 4. The non-transitory computer-readable medium of claim 3,wherein the computer-readable instructions further cause the computingdevice to: for each pair of training signals of the plurality of pairsof training signals, compute a conjugate of a first received signal ofthe pair of training signals to define a conjugate first receivedtraining signal; and compute a Kronecker product of the conjugate firstreceived training signal and a second received signal of the pair oftraining signals to define a differential measurement signal for thepair of training signals; compute a first estimated channel matrix fromthe defined differential measurement signal computed for each pair oftraining signals of the plurality of pairs of training signals; andcompute a second estimated channel matrix from the first estimatedchannel matrix; wherein the spatial filter matrix is computed from thesecond estimated channel matrix.
 5. The non-transitory computer-readablemedium of claim 4, wherein each pair of the plurality of pairs oftraining signals is selected so that a single column of the firstestimated channel matrix is estimated from the defined differentialmeasurement signal for the associated pair.
 6. The non-transitorycomputer-readable medium of claim 4, wherein the spatial filter matrixis computed using H_(o) ^(H) (ρH_(o)H_(o) ^(H)+σ²I_(n))⁻¹, where H_(o)is the second estimated channel matrix, H_(o) ^(H) is a hermitian matrixcomputed from H_(o), ρ is an estimated value of a signal to noise ratioof the plurality of pairs of training signals, σ² is an estimated noisepower matrix computed from the plurality of pairs of training signals,and I_(n) is an identity matrix.
 7. The non-transitory computer-readablemedium of claim 4, wherein computing the first estimated channel matrixcomprises solving z=H_(d)x, where H_(d) is the first estimated channelmatrix, z=vec(rr_(T) ^(H)), x=vec(ss_(T) ^(H)), where r is the secondreceived signal of the pair of training signals, r_(T) ^(H) is ahermitian transpose of the first received signal of the pair of trainingsignals, where s is a second transmitted signal of the pair of trainingsignals, and s_(T) ^(H) is a hermitian transpose of a first transmittedsignal of the pair of training signals.
 8. The non-transitorycomputer-readable medium of claim 7, wherein computing the secondestimated channel matrix comprises: define a column matrix index as one;define a row matrix index as one; define a second column matrix index asone; define a number of antennas as a number of the second plurality ofantennas; (a) select n column elements from H_(d)(r_(i), c₁), where n isthe number of antennas, c₁ is the column matrix index, and r_(i)=r, . .. , r+n−1, where r is the row matrix index; (b) select a column valuefrom H_(d)(c₁, c₁); (c) normalize each of the selected n column elementsusing a square root of the selected column value; (d) store thenormalized n column elements in H_(o)(r_(j), c₂), where H_(o) is thesecond estimated channel matrix, c₂ is the second column matrix index,and r_(j)=1, . . . , n; (e) increment the row matrix index using r=r+n;(f) increment the column matrix index using c₁=c₁+n+1; (g) increment thesecond column matrix index using c₂=c₂+1; and (h) repeat (a) to (g)until the second column matrix index is greater than the number ofantennas.
 9. A receiver comprising: a processor; and a non-transitorycomputer-readable medium operably coupled to the processor, thecomputer-readable medium having computer-readable instructions storedthereon that, when executed by the processor, cause the receiver tospatially filter a first signal using a spatial filter matrix to definea first filtered signal, wherein the first signal is a result of a firsttransmitted signal transmitted by a first plurality of antennas andreceived by a second plurality of antennas connected to the receiver;compute a conjugate of the first filtered signal to define a conjugatefirst signal; spatially filter a second signal using the spatial filtermatrix to define a second filtered signal, wherein the second signal isa result of a second transmitted signal transmitted by the firstplurality of antennas and received by the second plurality of antennasconnected to the receiver, wherein the second signal is received afterthe first signal; compute a Hadamard product of the first filteredsignal and the conjugate first signal to define a differentialmeasurement signal; and detect a plurality of information symbols fromthe differential measurement signal.
 10. The receiver of claim 9,wherein the spatial filter matrix is computed from an estimated channelmatrix.
 11. The receiver of claim 9, wherein the estimated channelmatrix is computed by transmitting a plurality of pairs of trainingsignals before the first signal is received.
 12. The receiver of claim11, wherein the computer-readable instructions further cause thereceiver to: for each pair of training signals of the plurality of pairsof training signals, compute a conjugate of a first received signal ofthe pair of training signals to define a conjugate first receivedtraining signal; and compute a Kronecker product of the conjugate firstreceived training signal and a second received signal of the pair oftraining signals to define a differential measurement signal for thepair of training signals; compute a first estimated channel matrix fromthe defined differential measurement signal computed for each pair oftraining signals of the plurality of pairs of training signals; andcompute a second estimated channel matrix from the first estimatedchannel matrix; wherein the spatial filter matrix is computed from thesecond estimated channel matrix.
 13. The receiver of claim 12, whereineach pair of the plurality of pairs of training signals is selected sothat a single column of the first estimated channel matrix is estimatedfrom the defined differential measurement signal for the associatedpair.
 14. The receiver of claim 12, wherein the spatial filter matrix iscomputed using H_(o) ^(H)(ρH_(o)H_(o) ^(H)+σ²I_(n))⁻¹, where H_(o) isthe second estimated channel matrix, H_(o) ^(H) is a hermitian matrixcomputed from H_(o), ρ is an estimated value of a signal to noise ratioof the plurality of pairs of training signals, σ² is an estimated noisepower matrix computed from the plurality of pairs of training signals,and I_(n) is an identity matrix.
 15. The receiver of claim 12, whereincomputing the first estimated channel matrix comprises solving z=H_(d)x,where H_(d) is the first estimated channel matrix, z=vec(rr_(T) ^(H)),x=vec(ss_(T) ^(H)), where r is the second received signal of the pair oftraining signals, r_(T) ^(h) is a hermitian transpose of the firstreceived signal of the pair of training signals, where s is a secondtransmitted signal of the pair of training signals, and s_(T) ^(H) is ahermitian transpose of a first transmitted signal of the pair oftraining signals.
 16. The receiver of claim 15, wherein computing thesecond estimated channel matrix comprises: define a column matrix indexas one; define a row matrix index as one; define a second column matrixindex as one; define a number of antennas as a number of the secondplurality of antennas; (a) select n column elements from H_(d)(r_(i),c₁), where n is the number of antennas, c₁ is the column matrix index,and r_(i)=r, . . . , r+n−1, where r is the row matrix index; (b) selecta column value from H_(d)(c₁, c₁); (c) normalize each of the selected ncolumn elements using a square root of the selected column value; (d)store the normalized n column elements in H_(o)(r_(j), c₂), where H_(o)is the second estimated channel matrix, c₂ is the second column matrixindex, and r_(j)=1, . . . , n; (e) increment the row matrix index usingr=r+n; (f) increment the column matrix index using c₁=c₁+n+1; (g)increment the second column matrix index using c₂=c₂+1; and (h) repeat(a) to (g) until the second column matrix index is greater than thenumber of antennas.
 17. A method of detecting a plurality of informationsymbols, the method comprising: spatially filtering, by a receiver, afirst signal using a spatial filter matrix to define a first filteredsignal, wherein the first signal is a result of a first transmittedsignal transmitted by a first plurality of antennas and received by asecond plurality of antennas connected to the receiver; computing, bythe receiver, a conjugate of the first filtered signal to define aconjugate first signal; spatially filtering, by the receiver, a secondsignal using the spatial filter matrix to define a second filteredsignal, wherein the second signal is a result of a second transmittedsignal transmitted by the first plurality of antennas and received bythe second plurality of antennas connected to the receiver, wherein thesecond signal is received after the first signal; computing, by thereceiver, a Hadamard product of the first filtered signal and theconjugate first signal to define a differential measurement signal; anddetecting, by the receiver, a plurality of information symbols from thedifferential measurement signal.
 18. The method of claim 17, wherein thespatial filter matrix is computed from an estimated channel matrix. 19.The method of claim 17, wherein the estimated channel matrix is computedby transmitting a plurality of pairs of training signals before thefirst signal is received.
 20. The method of claim 19, furthercomprising: for each pair of training signals of the plurality of pairsof training signals, computing, by the receiver, a conjugate of a firstreceived signal of the pair of training signals to define a conjugatefirst received training signal; and computing, by the receiver, aKronecker product of the conjugate first received training signal and asecond received signal of the pair of training signals to define adifferential measurement signal for the pair of training signals;computing, by the receiver, a first estimated channel matrix from thedefined differential measurement signal computed for each pair oftraining signals of the plurality of pairs of training signals; andcomputing, by the receiver, a second estimated channel matrix from thefirst estimated channel matrix; wherein the spatial filter matrix iscomputed from the second estimated channel matrix.